What is the integral from 0 to 1 of xcos(xy) ?

The integral from 0 to 1 of xcos(xy) is sin(x)

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integral from 0 to 1 of xcos(xy)

\int_{0}^{1} x cos (x y) d y

sin (x)

Solution steps

\int_{0}^{1} x cos (x y) d y

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= x \cdot \int_{0}^{1} cos (x y) d y

Apply u-substitution

: \int_{0}^{x} \frac{cos ( u )}{x} d u

= x \cdot \int_{0}^{x} \frac{cos ( u )}{x} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= x \frac{1}{x} \cdot \int_{0}^{x} cos (u) d u

Use the common integral:

\int cos (u) d u = sin (u)

= x \frac{1}{x} [sin (u)]_{0}^{x}

Simplify

x \frac{1}{x} [sin (u)]_{0}^{x} : [sin (u)]_{0}^{x}

= [sin (u)]_{0}^{x}

Compute the boundaries:

sin (x)

= sin (x)

\int_{3}^{5} 3 x^{2} + 2 x d x

\int_{2}^{2} \frac{1}{x} d x

\int_{1}^{13} 6 x^{2} + 3 x - 2 d x

\int_{1}^{3} (x^{- 2} - \frac{3}{x} + 2 x) d x

\int_{3}^{3} \frac{x - 2}{x + 2} d x

What is the integral from 0 to 1 of xcos(xy) ?
The integral from 0 to 1 of xcos(xy) is sin(x)